Finite Group Databasebeta

← #366463

Subgroups — #366463

Order 512.

1018 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21145C211yesmaximal, characteristic, complemented
2#20645C211yesmaximal, characteristic
3#20629C211yesmaximal, characteristic
4C2×#1768C211yesmaximal, characteristic, complemented
5C2×#1759C211yesmaximal, characteristic
6#11317C211yesmaximal, characteristic, complemented
7#11317C211yesmaximal, characteristic, complemented
8#11068C211yesmaximal, characteristic
9#11068C211yesmaximal, characteristic
10#10930C211yesmaximal, characteristic
11#10929C211yesmaximal, characteristic
12#10883C211yesmaximal, characteristic, complemented
13#10882C211yesmaximal, characteristic, complemented
14#10853C211yesmaximal, characteristic, complemented
15#10840C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#408C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29C22×#97C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31#2285C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1940—22no—
34#1939—22no—
35#1805C412yescomplemented
36#1804C412yescomplemented
37#1803C412yescomplemented
38#1802C412yescomplemented
39#1798—22no—
40#1797—22no—
41#1768—24no—
42#1759—24no—
43C2×#341C2211yescharacteristic
44C2×#341C2211yescharacteristic
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47#1735C2211yescharacteristic
48#1735C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55#1729C2211yescharacteristic
56#1729C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C22×#41C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C4×#41—22no—
71C4×#41—22no—
72C4×#41—22no—
73C4×#41—22no—
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4C2×C411yescharacteristic
80C2×#117—21no—
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C2×#117C2311yescharacteristic
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#424—22no—
89#424—22no—
90#424—41no—
91#424—41no—
92#423—22no—
93#423—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8C2×C412yes—
103C2×C4⋊C8C2×C412yes—
104C2×C4⋊C8D412yes—
105C2×C4⋊C8D412yes—
106C2×#104—22no—
107C2×#104—22no—
108C2×#104—22no—
109C2×#104—22no—
110C2×#104—22no—
111C2×#104Q814yes—
112#416—41no—
113#416—41no—
114#416—41no—
115#416—41no—
116C2×#103C2×C412yes—
117C2×#103C2×C412yes—
118C2×#102C2×C412yes—
119C2×#102C2×C412yes—
120#408—22no—
121#408—22no—
122#408—41no—
123#408—41no—
124C2×C22⋊C8—22no—
125C2×C22⋊C8—22no—
126C4×M16—22no—
127C4×M16—22no—
128C2×#97—22no—
129C2×#97—21no—
130C2×#97—41no—
131C2×#97—41no—
132C2×C4×C8—22no—
133C2×C4×C8—21no—
134C2×C4×C8—41no—
135C2×C4×C8—41no—
136#384C2×C412yes—
137#382C2×C412yes—
138#380—21no—
139#380—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142#376—22no—
143#376D414yes—
144C2×#95—21no—
145C2×#95—21no—
146#341—41no—
147#341—41no—
148#341—41no—
149#341—41no—
150#340C2×C412yes—
151#340C2×C412yes—
152#340C2×C412yes—
153#340C2×C412yes—
154#339—22no—
155#339—22no—
156#339—22no—
157#339—22no—
158#339—41no—
159#339—41no—
160#336—22no—
161#336—22no—
162#336—22no—
163#336—22no—
164#336—22no—
165#336—22no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41C2×Q812yes—
177C2×#41#4112yes—
178C22×Q8C22×C411yescharacteristic
179C22×D4—21no—
180C22×D4C22×C411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4#4111yescharacteristic
187C23×C4#4111yescharacteristic
188C23×C4C2411yescharacteristic
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16C2×Q812yes—
212C2×M16#4112yes—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8C22×C412yes—
242C22×C8C22×C412yes—
243C22×C8C2×D412yes—
244C22×C8C2×D412yes—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—42no—
254C4×D4—42no—
255C4×D4—42no—
256C4×D4—42no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—42no—
270#117—42no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C4⋊C4C22×C411yescharacteristic
277C2×C4⋊C4C22×C412yes—
278C2×C4⋊C4C22×C412yes—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42C22×C411yescharacteristic
300C2×C42C22×C411yescharacteristic
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×Q812yes—
307C2×C42#4112yes—
308C2×C42#4112yes—
309C2×C42#4112yes—
310C2×C42#4112yes—
311C4⋊C8—24no—
312C4⋊C8—24no—
313C4⋊C8—24no—
314C4⋊C8—24no—
315C4⋊C8—44no—
316C4⋊C8—44no—
317C4⋊C8—42no—
318C4⋊C8—42no—
319#104—28no—
320#104—28no—
321#103—42no—
322#103—42no—
323#102—42no—
324#102—42no—
325C22⋊C8—44no—
326C22⋊C8—44no—
327#97—44no—
328#97—42no—
329C4×C8—44no—
330C4×C8—42no—
331#95—22no—
332#95—22no—
333#95—24no—
334#95—24no—
335C24—21no—
336C24C2×#4111yescharacteristic
337#41—22no—
338#41—22no—
339#41—22no—
340#41—22no—
341#41—22no—
342#41—22no—
343#41—22no—
344#41—22no—
345#41—41no—
346#41—41no—
347#41—41no—
348#41—41no—
349#41—41no—
350#41—41no—
351#41—41no—
352#41—41no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8C4×Q812yes—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4C4×Q812yes—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—82no—
398C22×C4C4×D411yescharacteristic
399C22×C4C4×D411yescharacteristic
400C22×C4C4×D411yescharacteristic
401C22×C4C4×D411yescharacteristic
402C22×C4#12312yes—
403C22×C4#12312yes—
404C22×C4#12612yes—
405C22×C4#12612yes—
406C22×C4C23×C411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×Q811yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C4H(5,𝔽2)11yescharacteristic
418C22×C4H(5,𝔽2)11yescharacteristic
419C22×C42−1+411yescharacteristic
420M16—22no—
421M16—22no—
422M16—22no—
423M16—22no—
424M16—41no—
425M16—41no—
426M16—41no—
427M16—41no—
428M16—41no—
429M16—41no—
430M16—41no—
431M16—41no—
432M16—41no—
433M16—41no—
434M16—41no—
435M16—41no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—82no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—42no—
510C42—42no—
511C42#11712yes—
512C42#11712yes—
513C42C4×D412yes—
514C42C4×D412yes—
515C42#12112yes—
516C42#12112yes—
517C42C22⋊Q814yes—
518C42#12412yes—
519C42#12412yes—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23C2×C4×D411yescharacteristic
528C23C2×#12311yescharacteristic
529C23C2×#12611yescharacteristic
530C23#55211yescharacteristic
531C23#55311yescharacteristic
532C23#55411yescharacteristic
533C23#55511yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—41no—
537Q8—41no—
538D4—22no—
539D4—22no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—41no—
545D4—41no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4#38812yes—
584C2×C4#38812yes—
585C2×C4#38812yes—
586C2×C4#38812yes—
587C2×C4C2×#11711yescharacteristic
588C2×C4C2×C4×D411yescharacteristic
589C2×C4C2×C4×Q811yescharacteristic
590C2×C4C4×#4111yescharacteristic
591C2×C4#51811yescharacteristic
592C2×C4#52011yescharacteristic
593C2×C4C2×#12111yescharacteristic
594C2×C4C2×C22⋊Q811yescharacteristic
595C2×C4#52511yescharacteristic
596C2×C4C2×#12411yescharacteristic
597C2×C4#52911yescharacteristic
598C2×C4#53311yescharacteristic
599C2×C4#53511yescharacteristic
600C2×C4#53711yescharacteristic
601C2×C4#54011yescharacteristic
602C2×C4#54211yescharacteristic
603C2×C4#54611yescharacteristic
604C2×C4#54611yescharacteristic
605C2×C4D4×Q811yescharacteristic
606C2×C4#55011yescharacteristic
607C2×C4#55311yescharacteristic
608C2×C4#55311yescharacteristic
609C2×C4#55311yescharacteristic
610C2×C4#55311yescharacteristic
611C2×C4#55411yescharacteristic
612C2×C4#55411yescharacteristic
613C2×C4#55611yescharacteristic
614C2×C4#55611yescharacteristic
615C8—42no—
616C8—42no—
617C8—42no—
618C8—42no—
619C8—42no—
620C8—42no—
621C8—81no—
622C8—81no—
623C22—22no—
624C22—22no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—22no—
630C22—22no—
631C22—42no—
632C22—44no—
633C22—82no—
634C22—81no—
635C22—81no—
636C22C2×#38811yescharacteristic
637C22#236011yescharacteristic
638C22#236311yescharacteristic
639C22#248511yescharacteristic
640C22#248711yescharacteristic
641C22#249411yescharacteristic
642C22#249711yescharacteristic
643C22#295512yes—
644C22#295712yes—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—44no—
650C4—42no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—81no—
655C4—81no—
656C4C4×C22⋊Q811yescharacteristic
657C4#232411yescharacteristic
658C4#234511yescharacteristic
659C4#234711yescharacteristic
660C4#249011yescharacteristic
661C4#249211yescharacteristic
662C4#253511yescharacteristic
663C4#253811yescharacteristic
664C2—21no—
665C2—21no—
666C2—21no—
667C2—21no—
668C2—82no—
669C2#1399311yescharacteristic
670C2#2109811yescharacteristic
671C2#2110011yescharacteristic
672C2#2298412yes—
673C2#2311512yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21145C2semidirect
4. C2×#1768C2semidirect
6. #11317C2semidirect
7. #11317C2semidirect
12. #10883C2semidirect
13. #10882C2semidirect
14. #10853C2semidirect
15. #10840C2semidirect
35. #1805C4semidirect
36. #1804C4semidirect
37. #1803C4semidirect
38. #1802C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#211452
#206452
#206292
C2×#17682
C2×#17592
#113172
#110682
#109302
#109292
#108832
#108822
#108532
#108402